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| Mirrors > Home > MPE Home > Th. List > fmptapd | Structured version Visualization version GIF version | ||
| Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.) |
| Ref | Expression |
|---|---|
| fmptapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| fmptapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| fmptapd.s | ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) |
| fmptapd.c | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) |
| Ref | Expression |
|---|---|
| fmptapd | ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fmptapd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) | |
| 2 | fmptapd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 3 | fmptapd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 4 | 1, 2, 3 | fmptsnd 7167 | . . 3 ⊢ (𝜑 → {〈𝐴, 𝐵〉} = (𝑥 ∈ {𝐴} ↦ 𝐶)) |
| 5 | 4 | uneq2d 4121 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
| 6 | mptun 6681 | . . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
| 7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
| 8 | fmptapd.s | . . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) | |
| 9 | 8 | mpteq1d 5200 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
| 10 | 5, 7, 9 | 3eqtr2d 2803 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∪ cun 3902 {csn 4588 〈cop 4594 ↦ cmpt 5191 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-pr 5403 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-opab 5173 df-mpt 5192 |
| This theorem is used by: fmptpr 7170 poimirlem3 38302 poimirlem4 38303 poimirlem16 38315 poimirlem17 38316 poimirlem19 38318 poimirlem20 38319 |
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